To model subsurface flow in uncertain heterogeneous\ fractured media an elliptic equation with a discontinuous stochastic diffusion coefficient - also called random field - may be used. In case of a one-dimensional parameter space, Lévy processes allow for jumps and display great flexibility in the distributions used. However, in various situations (e.g. microstructure modeling), a one-dimensional parameter space is not sufficient. Classical extensions of Lévy processes on two parameter dimensions suffer from the fact that they do not allow for spatial discontinuities. In this paper a new subordination approach is employed to generate Lévy-type discontinuous random fields on a two-dimensional spatial parameter domain. Existence and uniqueness of a (pathwise) solution to a general elliptic partial differential equation is proved and an approximation theory for the diffusion coefficient and the corresponding solution provided. Further, numerical examples using a Monte Carlo approach on a Finite Element discretization validate our theoretical results.
General elliptic equations with spatially discontinuous diffusion coefficients may be used as a simplified model for subsurface flow in heterogeneous or fractured porous media. In such a model, data sparsity and measurement errors are often taken into account by a randomization of the diffusion coefficient of the elliptic equation which reveals the necessity of the construction of flexible, spatially discontinuous random fields. Subordinated Gaussian random fields are random functions on higher dimensional parameter domains with discontinuous sample paths and great distributional flexibility. …
Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear elliptic PDEs with Lipschitz nonlinearities and to prove that the proposed approximation schemes overcome the curse of dimensionality in the numerical approximation of such semilinear e…
Mesh sensitivity of finite element solution for linear elliptic partial differential equations is analyzed. A bound for the change in the finite element solution is obtained in terms of the mesh deformation and its gradient. The bound shows how the finite element solution changes continuously with the mesh. The result holds in any dimension and for arbitrary unstructured simplicial meshes, general linear elliptic partial differential equations, and general finite element approximations.
Elliptic partial differential equations (PDEs) arise in many areas of computational sciences such as computational fluid dynamics, biophysics, engineering, geophysics and more. They are difficult to solve due to their global nature and sometimes ill-conditioned operators. We review common discretization methods for elliptic PDEs such as the finite difference, finite volume, finite element, and spectral methods and the linear systems they form. We also provide an overview of classic to modern solution methods for the linear systems formed by these discretization methods. These methods include s…
Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting. Extensions to other variants of adaptive collocation methods (including the classical one proposed in the paper "Dimension-adaptive tensor-product quadratuture" Computing (2003) …
Almost all materials are anisotropic. In this paper, interface relations of anisotropic elliptic partial differential equations involving discontinuities across interfaces are derived in two and three dimensions. Compared with isotropic cases, the invariance of partial differential equations and the jump conditions under orthogonal coordinates transformation is not valid anymore. A systematic approach to derive the interface relations is established in this paper for anisotropic elliptic interface problems, which can be important for deriving high order accurate numerical methods.
The Virtual Element Method (VEM) is a very effective framework to design numerical approximations with high global regularity to the solutions of elliptic partial differential equations. In this paper, we review the construction of such approximations for an elliptic problem of order $p_1$ using conforming, finite dimensional subspaces of $ H^{p_2}(Ω)$, where $p_1$ and $p_2$ are two integer numbers such that $p_2 \geq p_1 \geq 1$ and $Ω\in R^2$ is the computational domain. An abstract convergence result is presented in a suitably defined energy norm. The space formulation and major aspects suc…
An economic cascadic tensor multigrid method for solving high dimensional elliptic linear partial differential problems
Guilin University of Electronic Technology（桂林电子科技大学）；Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation（广西高校数据分析与计算重点实验室）；Center for Applied Mathematics of Guangxi (GUET)（广西应用数学中心）
This paper proposes a method for rigorously analyzing the sign-change structure of solutions of elliptic partial differential equations subject to one of the three types of homogeneous boundary conditions: Dirichlet, Neumann, and mixed. Given explicitly estimated error bounds between an exact solution $ u $ and a numerically computed approximate solution $ \hat{u} $, we evaluate the number of sign-changes of $ u $ (the number of nodal domains) and determine the location of zero level-sets of $ u $ (the location of the nodal line). We apply this method to the Dirichlet problem of the Allen-Cahn…
The Virtual Element Method is well suited to the formulation of arbitrarily regular Galerkin approximations of elliptic partial differential equations of order $2p_1$, for any integer $p_1\geq 1$. In fact, the virtual element paradigm provides a very effective design framework for conforming, finite dimensional subspaces of $H^{p_2}(Ω)$, $Ω$ being the computational domain and $p_2\geq p_1$ another suitable integer number. In this study, we first present an abstract setting for such highly regular approximations and discuss the mathematical details of how we can build conforming approximation s…
We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized {nonlinear} elliptic partial differential equations (PDEs). CB-pMOR is designed to deal with large-scale problems for which full-order solves are not affordable in a reasonable time frame or parameters' variations induce topology changes that prevent the application of monolithic pMOR techniques. We rely on the partition-of-unity method (PUM) to devise global approximation spaces from local reduced spaces, and on Galerkin projection to compute the global state estimate. We propose a randomiz…
University of Oxford（牛津大学）；Cornell University（康奈尔大学）
Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning the associated Green's function $G$. By exploiting the hierarchical low-rank structure of $G$, we show that one can construct an approximant to $G$ that converges almost surely and achieves a relative error of $\mathcal{O}(Γ_ε^{-1/2}\log^3(1/ε)ε)$ using at most $\mathcal{O}(ε^{-6}\log^4(1/ε))$ input-output training pairs with high probability, for any $0<ε<1$. The quantity $0<Γ_ε\leq 1$ characterizes the quality of the training dataset.…
